(a) Evaluate \(\int_{1} ^{2} \frac{x}{\sqrt{5 - x^{2}}} \mathrm {d} x\)

(b)(i) Evaluate: \(\begin{vmatrix} 2 & -3 & 1 \\ 0 & 1 & -2 \\ 1 & 2 & -3 \end{vmatrix}\)

(ii) Using your answer in b(i), solve the simultaneous equations :

\(2x - 3y + z = 10\)

\(y - 2z = -7\)

\(x + 2y - 3z = -9\)

(a) Use the trapezium rule with five ordinates to evaluate \(\int_{0} ^{1} \frac{3}{1 + x^{2}} \mathrm {d} x\), correct to four significant figures.

(b) If \(A = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}\), find the image of the point (1, 2) under the linear transformation \(A^{2} + A + 2I\), where I is the \(2 \times 2\) unit matrix.

(a) Simplify \(^{n + 1}C_{3} - ^{n - 1}C_{3}\)

(b) A fair die is thrown five times. Calculate, correct to three decimal places, the probability of obtaining (i) at most two sixes ; (ii) exactly three sixes.

(a) The distribution of the lives (in days) of 40 transitor batteries is shown in the table:

(a) Draw a histogram for the distribution.

(b) Use your graph in (a) to determine the mode for the distribution.

(c) Using an assumed mean of 43 days, calculate the mean of the distribution.

(d) What percentage number of batteries will live for less than 31 days or more than 45 days?

The table below shows the corresponding values of two variables X and Y.

(a) Plot a scatter diagram to represent the data.

(b) Calculate \(\bar{x}\), the mean of X and \(\bar{y}\), the mean of Y.

(c) Draw the line of best fit to pass through \((\bar{x}, \bar{y})\).

(d) From your graph in (c), determine the (i) relationship between X and Y ; (ii) value of Y when X is 24.